Basic Probability Theory Question about Lebesgue measure

Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable. Is there a reason why the choice is also preferred in physics?

Pikkugnome said:

Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable.

Non-measurable sets are fairly pathological.

Pikkugnome said:

Is there a reason why the choice is also preferred in physics?

Sets that are relevant to physical phenomena are generally measureable.

PeroK said:

Non-measurable sets are fairly pathological.

Good point. But I wonder if they might actually be more numerous than measurable sets, like the transcendental numbers versus the algebraic numbers.

PeroK said:

Sets that are relevant to physical phenomena are generally measureable.

Can you think of any non-measurable set that would be of interest in physics? I can't.

FactChecker said:

Can you think of any non-measurable set that would be of interest in physics? I can't.

I think the question was asked on here a few years ago, in a slightly different context. To obtain a subset of ##\mathbb R## that is not Borel-measurable requires the axiom of choice. Is the axiom of choice ever relevant to mathematical physics?

Alternatively, we could abondon ZFC and study mathematics where the AC fails and every set is Lebesque measurable. That was suggested to me in 1984 as a possible postgraduate research project!

PeroK said:

Is the axiom of choice ever relevant to mathematical physics?

If so, I will definitely reconsider Banach-Tarski.

The topic is measurable sets in the mathematical sense. A perhaps related topic is measurable sets in the laboratory sense. For example, if Nature produces an outcome from a continuous curve of possibilities, it's usually only possible to measure this outcome with finite precision. So if the model for a practical laboratory measurement of a continuous quantity is an interval or some sort of probability distribution, can there be any non-measureable (in the mathematical sense) sets composed of such measurements?

If Banach Tarski could be physically realizable, diamonds, gold, would be worthless.

Pikkugnome said:

Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable. Is there a reason why the choice is also preferred in physics?

Because measurable sets have the properties we believe any physically measurable things should have. These are encoded in the properties of a sigma algebra. If you extended probabilities to non-measurable sets you would open the door to a whole set of paradoxes.

WWGD said:

If Banach Tarski could be physically realizable, diamonds, gold, would be worthless.

I remember a colloquium about the proof of Banach-Tarski. The referent argued that it is less the AC that is against our intuition, rather it is our concept of a point that lacks any physical reality. This is an interesting point of view since it is primarily AC that is considered the culprit. But the more I think about it the more I have to agree to that professor whose name I have unfortunately forgotten.

WWGD said:

If Banach Tarski could be physically realizable, diamonds, gold, would be worthless.

I can just see Marilyn Monroe singing "Banach-Tarski is a girl's best friend"!

PeroK said:

I can just see Marilyn Monroe singing "Banach-Tarski is a girl's best friend"!

And Marylin Idiot Savant is Probability/Mathematics ' biggest enemy *

* Ignoring for now Archie Plutonium.

fresh_42 said:

I remember a colloquium about the proof of Banach-Tarski. The referent argued that it is less the AC that is against our intuition, rather it is our concept of a point that lacks any physical reality. This is an interesting point of view since it is primarily AC that is considered the culprit. But the more I think about it the more I have to agree to that professor whose name I have unfortunately forgotten.

You wanted to write down their name, but the margin was too..

fresh_42 said:

I remember a colloquium about the proof of Banach-Tarski. The referent argued that it is less the AC that is against our intuition, rather it is our concept of a point that lacks any physical reality. This is an interesting point of view since it is primarily AC that is considered the culprit. But the more I think about it the more I have to agree to that professor whose name I have unfortunately forgotten.

There's also the magic on how that collection of discrete , finite, points magically turns into a continuum, with nonzero length, area, etc.

WWGD said:

There's also the magic on how that collection of discrete , finite, points magically turns into a continuum, with nonzero length, area, etc.

The subject of this thread is somehow reached again. We ignore points since they have no positive Lebesgues measure. But without them, we wouldn't have science as we use it today. I don't think we would get very far with only three-dimensional objects.

fresh_42 said:

The subject of this thread is somehow reached again. We ignore points since they have no positive Lebesgues measure. But without them, we wouldn't have science as we use it today. I don't think we would get very far with only three-dimensional objects.

The same applies for all dimensions. And for length/area, etc.